Studying the Complexity of Rainfall Within California Via a Fractal Geometric Method

Abstract

A deterministic geometric approach, the fractal–multifractal (FM) method, useful in modeling storm events and recently adapted in order to encode highly intermittent daily rainfall records, is employed to study the complexity of rainfall sets within California. Specifically, sets—from south to north—at Cherry Valley, Merced, Sacramento and Shasta Dam and containing, respectively 59, 116, 115, and 72 years, all ending at water year 2015, are studied. The analysis reveals that: (a) the FM approach provides faithful encodings of all records, by years, with mean square and maximum errors in accumulated rain that are less than a mere 2 and 10%, respectively; (b) the evolution of the corresponding “best” FM parameters, allowing visualization of the inter-annual rainfall dynamics from a reduced vantage point, exhibit a highly entropic variation that prevents discriminating between sites and extrapolating to the future; and (c) the rain signals at all sites may be termed “equally complex,” as usage of k-means clustering and conventional phase-space analysis of FM parameters yields comparable results for all sites.